Eight minutes into the lesson, the class reconvened. Each of the five groups sent a representative to the front, one by one, to present and discuss a problem from the test with the class. Let us look at an excerpt from the transcript to get a sense of what the class discussion was like. Here, for example, the student, at the front of the room is presenting problem number 16: "Factor completely, by first factoring out the greatest common factor and then factoring the resulting polynomial: 8x2 + 8:' We present a somewhat lengthy excerpt.

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This excerpt is quite typical of the whole lesson. On the tape we see a class struggling to attach words to their understandings and their solution methods, and the teacher constantly acting to facilitate the exchange. The last ten minutes of the lesson were spent on a new problem that involved finding the slope of the line crossing through two points. The slope is negative, which some of the students find confusing. Again, there is a lively discussion as the teacher tries to mediate between two different solution methods, only one of which ends up being correct.

This teacher explained that the lesson was not in accord with current ideas because "current mathematical thinking is that factoring is not an important concept." She then added the comment: "I disagree' Chapter 7. Discussion and Conclusions We have taken numerous measurements and found numerous differences among German, Japanese, and U.S. lessons. But what do they all mean? Which of these indicators are important, and which not?

How can we put them all together to describe a German math lesson, a Japanese math lesson, and a U.S. math lesson?

The real answer is that we cannot fully put them together, at least not yet. We are at the beginning stages of this inquiry. Although we are beginning to understand the characteristics of the indicators themselves, we still have much to do to understand how these indicators relate to the underlying models of instruction that govern their performance. Nevertheless, we can give our impressions, our speculations, about the models of instruction that produce the findings presented in this report. What follows, therefore, should be taken as preliminary and speculative.

TYPICAL LESSONS: GERMANY, JAPAN, AND THE UNITED STATES

Throughout this report we have used "lesson" as the unit, both of sampling and analysis. Although we believe there is validity in this approach, it also is important to consider that lessons, for teachers in all three cultures, are related to each other in sequences that form units of instruction across the school year. In our attempts to describe lessons, we necessarily must take this fact into account. Not all the parts of a typical lesson in a given culture, for example, will appear at all points in the course of a unit of instruction. Early lessons will include more development, later ones more practice. Because of our methodology, we cannot take relations among lessons into account.

With this limitation in mind, we nevertheless will proceed to construct, based on the data, typical lessons in Germany, Japan, and the United States. By combining a number of the indicators we have reported and by using the country comparisons as foils against which we can get sharper images of instructional patterns, we can begin to piece together descriptions of how "typical" lessons look in each country. The data presented in this report allow us to describe some characteristics of typical lessons.

For ease of reading, we present the typical lessons in as simple and straightforward way as possible.

This means that we have overgeneralized and have omitted qualifiers and caveats that indicate the tentativeness with which these stories should be interpreted. Nevertheless, the primary features of the stories are supported by the coded data presented earlier.

Germany The mathematics teacher sets the goal for the lesson as the acquisition of a skill or procedure for solving a mathematical problem (figure 12). It is likely that the particular skill or procedure the teacher intends to teach is relatively challenging mathematically (figure 10). She probably intends for the students to understand the rationale for the procedure, why it works (figure 18).

To achieve these goals, the teacher organizes the lesson so that most of the mathematical work during the lessons is done as a whole class (figure 39). The teacher does not lecture much to the students (figure 47); instead, she guides students through the development of the procedure by asking students to orally fill in the relevant information (figures 71-74, 78). This is done by presenting the students with a task such as finding the solution set to two simultaneous linear equations in two unknowns. If the problem is a relatively new one, the teacher generally works the problem at the board, eliciting ideas and procedures from the class as work on the problem progresses. If the problem is one they have already been introduced to, a student might be called to the chalkboard to work the problem. The problem might be slightly different than problems students have worked before but the method to solve the problem has been introduced previously and applied in related situations. The class is expected to monitor the student's work, to catch errors that are made, and to help the student when he or she gets stuck.

The teacher keeps the student and class moving forward by asking questions about next steps and about why such steps are appropriate.

After two or three similar problems have been worked in this way, the teacher summarizes the activity by pointing to the principle or property (figure 24) that guides the deployment of the procedure in these new situations. For the remaining minutes of the class period, she assigns several problems in which students practice the procedure in similar situations (figure 65).

Japan The goal the teacher sets for the lesson is for students to develop mathematical thinking (figure 12) rather than to acquire a particular mathematical procedure as in other countries. Planning for the lesson involves selecting challenging mathematical problems (figure 10) that might involve the development of several student-presented methods of solution (figure 22) or the development of a mathematical proof (see chapter 3 section "Proofs").

The lesson begins with the teacher posing the selected problem. The students are then asked to work on the problem at their seats (figures 39, 41) in order to generate a solution method for the problem (figures 33, 51), sometimes individually and sometimes in groups. During the seatwork period, the teacher circulates around the class, noting the different methods that students are constructing. She then reconvenes the class and asks particular students to come to the front and share their methods (figures 22, 54). Occasionally, the teacher provides a brief lecture (figure 47), pointing out a property or principle inherent in the method (figure 24) or explaining the advantage of particular methods that have been shared. The cycle of the teacher presenting a problem, students working on the problem at their seats, and students sharing their solutions with the class, is repeated several times during lesson (figure 38).

Much mathematical work in the lesson gets done during seatwork (figures 49, 51, 65). But students are given support and direction through the class discussion of the problem when it is posed (figure 50), through the summary explanations by the teacher (figure 47) after methods have been presented, through comments by the teacher that connect the current task with what students have studied in previous lessons or earlier in the same lesson (figure 80), and through the availability of a variety of mathematical materials and tools (figure 53).

United States The mathematics teacher sets the goal for the lesson as the acquisition of a skill or procedure for solving a mathematical problem (figure 12). The goal probably specifies a particular procedure that all students are to acquire rather than generating alternative solution methods (figure 22) and the emphasis will be placed on acquiring the mechanics of the procedure rather than learning why the procedure works (figures 18, 65). Compared to Germany and Japan, the procedure to be learned is at a relatively simple mathematical level (figure 10).

The lesson typically includes the teacher asking students about their homework (figure 46). The teacher works one or two problems with which students have had difficulty and the homework is collected. The teacher then presents one or more definitions or properties or principles (figure 24), often in the form of procedural rules, that will guide the students' application of the procedures for the next set of problems (figure 15). This might be in the form of a demonstration of a new procedure or a reminder of how a procedure is used in the situations presented in this lesson. Several examples are then worked together as a class (figures 38, 39), with the teacher using the chalkboard or the overhead projector (figures 52, 54). The problems are probably drawn from the textbook or a teacher-made worksheet (figure 53), and few materials or tools are used other than paper and pencil. The teacher guides the students through the procedure by asking short-answer questions of the students (figure 72), such as what is the partial result for this step in the procedure, or what, operation is called for in the next step.

The teacher assigns a number of problems of a similar kind (figures 33, 46) as homework. The students work on the problems until the end of the period. The problems function as practice exercises for the procedures demonstrated earlier (figures 51, 65).

COMPARING LESSON SCRIPTS

Comparing the lesson patterns or scripts across countries leads to some interesting observations. Both German and U.S. lessons follow what might be called an acquisition/application script (Hiebert et aL, During the acquisition phase, students are expected to learn how to solve particular types of 1996).

problems, often through watching demonstrations by the teacher or their peers. During the application phase, students are expected to practice what they have learned.

The acquisition part of the lesson occurs during classwork, the application part during seatwork. The role of the teacher during classwork is to lead students through an example and show them how to do it. The role of students is to pay attention, follow each step, respond to teachers' questions about next steps, and ask questions if they do not understand. In Germany, students often work demonstration examples at the chalkboard, as well, and serve as scribes for the class suggestions for how to work the problems. During seatwork, the role of teachers is to monitor students' work, giving help to students who are stuck or are making mistakes. The role of students is to practice procedures on similar problems.

Lessons in Japan look quite different. They follow what might be called a "problematizing" script (Hiebert et al., 1996). In this script, problem solving becomes the context in which competencies are simultaneously developed and utilized. This means that problems play a very different role in the lesson. Rather than problem solutions serving as the goal of the lesson, they are the means through which students are to understand properties or principles of mathematics. This difference in how problems are viewed leads to a difference in the way lessons are structured.

Instead of beginning with teacher-directed classwork followed by seatwork, the order is reversed.

Students try to solve problems on their own first, then comes a period of teacher-directed discussion.

In fact, there are often several of these cycles in one lesson. In the opening classwork segment, the teacher's role is to pose a task that students find problematic. In the following seatwork segment, the students' role is to develop a method for solving the problem. It is expected that students will struggle because they have not already acquired a procedure to solve the problem. The time spent struggling on one's own to work out a solution is considered an important part of the lesson. To complete the cycle, the class reconvenes to share the methods that have been developed.

1.53 In this script, the students are given more responsibility for doing the mathematical work. The teacher takes an active role in posing problems and helping students examine the advantages of different solution methods, but students are expected to struggle with mathematical problems and invent their own methods. By doing this, students may be engaged in different kinds of mathematical thinking than students in Germany or the United States. This means that the differences between lessons may be more than the quantitative differences shown in the figures; there may be qualitative differences in the kinds of mathematics in which students are engaged.

Although lessons in Germany and the United States seemed to follow a similar general script, there were some differences between them. As measured by international standards, the mathematical content of German lessons was significantly higher than that of U.S. lessons (figure 10). The content of German lessons was also structured in a more coherent way and focused to a greater extent on the development of concepts (rather than just the application of procedures; figure 18).

In fact, in the development of concepts, German lessons more closely resembled Japanese lessons than U.S. lessons. In Germany and Japan, concepts were usually developed over the course of the lesson.

Formulas, for example, were not just stated. They might be derived, or, at least, the rationale for them would be developed. In the United States, concepts were usually just stated. A formula would be presented without explaining why it worked.

The differences in cultural scripts that we have outlined can be summarized as follows (figure 86):

Figure 86 Comparison of steps typical of eighth-grade mathematics lessons in Japan, Germany, and the United States The emphasis on understanding is evident in the steps typical of Japanese

eighth-grade mathematics lessons:

Teacher poses a complex, thought-provoking problem.

Students struggle with the problem.

Various students present ideas or solutions to the class.

The teacher summarizes the class' conclusions.

Students practice similar problems.

In contrast, the emphasis on skill acquisition is evident in the steps

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